The present invention disclosed herein relates to methods for estimating statistical distribution characteristics of parameters used in developing products.
Qualities of products are basically dependent on design rules and processing conditions applied in designing and manufacturing of the products. With evolution of science and technology, procedures for designing and manufacturing industrial products have become more complicated. As a result, it is more difficult to analyze dependence of product quality relative to design rules and processing conditions. Considering that the accuracy and rapidity of such analysis operations contributes to shortening time-to-market for new products, it is necessary to analyze, correctly and rapidly, correlations among the design rules, the processing conditions, and the product qualities.
In further detail, manufacture of semiconductor integrated circuit by high technology processes is a typical one having complexity of design and fabrication process and difficulty in analyzing correlations according thereto. Usually, a manufacturer fabricates a semiconductor integrated circuit with reference to a specification defining required standards on electrical and structural characteristics. In the beginning of the semiconductor industry, circuit design verification according to the specification was conducted directly by a person, but now a computer system takes on the function of circuit design verification. Applying a computer system having an excellent computing power to the prior case had been quite successful. However, an operation speed and accuracy of the circuit design verification has been remarkably degraded as the integration density of semiconductor circuit has increased.
Further, as semiconductor devices become smaller in dimensions, a relative ratio of processing variations, which results from procedures of manufacturing such semiconductor devices, is increased. That is, if highly and lowly integrated semiconductor devices have a processing error of the same size, the highly integrated semiconductor device has the increased ratio of the size of processing error to the reference size or dimensions, compared with the lowly integrated semiconductor device. Therefore, in a procedure of designing a semiconductor integrated circuit, there is a rising need of considering even variations on manufacturing processes. Since variations in manufacturing processes affect a yield of a semiconductor device, it is important to estimate fluctuations on electrical characteristics of products according to the variations on manufacturing processes.
In detail, considering that electrical characteristics of semiconductor devices are subject to morphological/physical parameters (hereinafter, referred to as independent parameters) such as channel length (L), device width (W), doping profile (Na or Nd), oxide thickness (Tox), oxide permittivity (εox), and channel length modulation constant (λ), it is necessary to estimate statistical distributions of the independent parameters in order to enhance yields of the semiconductor devices. In a prior approach, as shown in FIG. 1, a predetermined process of simulation (S2) was carried out for estimating product characteristics. The process of simulation used input data with design data (i.e., the independent parameters) that was assumed to have normal distribution (S1). However, the normal distribution assumed for input data can be improper because of complicated reasons, such as the aforementioned variations on process. Incorrect input data results in inaccurate estimation to product characteristics, so it is insufficient to obtain a desired result just by assuming the distribution characteristics of design data, which are used as the input data, as being arranged in normal distribution profile. Thus, it is required to correctly estimate the product characteristics.
Nevertheless, it is generally not easy to estimate statistical distribution of the independent parameters. For instance, physical theories can be utilized to derive an equation defining correlations between the independent parameters and the dependent electrical characteristics, but those theoretical approaches are regarded as being successful only in very restrictive cases. In other words, as general cases, those equations are kinds of multi-variable functions. Further, since parameters of the equations are dependent on processing conditions (continuously updated for improving yields), it is mostly difficult to derive the equations through such theoretical approaches in practice. As a result, it is hard for the prior approach to obtain proper estimation results of statistical distribution characteristics for the independent parameters.
In other approaches to relieve such aforementioned difficulty, there are methods of estimating statistical distribution characteristics for the independent parameters by means of a model fitting process that needs a long arithmetic procedure. However, as those methods are based on the model fitting technique, they do not provide physical significance for correlations between the independent parameters and the electrical characteristics subject to the independent parameters, as well as taking a very long time to execute.